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<b>let </b><font color="Maroon" title="c1">W</font>, <font color="Maroon" title="c2">X</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>



<b>assume </b><a NAME="E1:36"/>
<font color="Maroon" title="c1">W</font> <a href="classes1.html#NR1" title="CLASSES1:NR.1">is_Tarski-Class</a> 
 ;<br/>

<a NAME="E2:36"/><b>then </b><i><font color="Green" title="E31">A1</font></i>: 
<font color="Maroon" title="c1">W</font> is <a href="classes1.html#V1" title="CLASSES1:attr.1">subset-closed</a>
 
<b>by </b><i><a class="ref" href="classes1.html#D2" title="CLASSES1:def.2">CLASSES1:def 2</a></i>;<br/>
<b>assume </b><a NAME="E3:36"/><i><font color="Green" title="E32">A2</font></i>: 
<font color="Maroon" title="c2">X</font> is <a href="ordinal1.html#V1" title="ORDINAL1:attr.1">epsilon-transitive</a>
 ;<br/>

<b>assume </b><a NAME="E4:36"/>
<font color="Maroon" title="c2">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">W</font>
 ;<br/>

<a NAME="E5:36"/><b>then </b>
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c2">X</font> <a href="card_1.html#NR2" title="CARD_1:NR.2">&lt;`</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c1">W</font>
 
<b>by </b><i><a class="txt" href="classes2.html#E2:36"><i><font color="Green" title="E31">A1</font></i></a>, <a class="ref" href="classes2.html#T1" target="_self" title="CLASSES2:th.1">Th1</a></i>;<br/>
<a NAME="E6:36"/><b>then </b>
(  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><a href="classes1.html#K6" title="CLASSES1:func.6">the_rank_of</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="card_1.html#NR2" title="CARD_1:NR.2">&lt;`</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c1">W</font> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c1">W</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c1">W</font> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c1">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c1">W</font> )
 
<b>by </b><i><a class="txt" href="classes2.html#E3:36"><i><font color="Green" title="E32">A2</font></i></a>, <a class="ref" href="classes2.html#T42" target="_self" title="CLASSES2:th.42">Th42</a>, <a class="ref" href="ordinal1.html#T22" title="ORDINAL1:th.22">ORDINAL1:22</a></i>;<br/>
<a NAME="E7:36"/><b>then </b>
 <a href="classes1.html#K6" title="CLASSES1:func.6">the_rank_of</a> <font color="Maroon" title="c2">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c1">W</font>
 
<b>by </b><i><a class="ref" href="card_3.html#T59" title="CARD_3:th.59">CARD_3:59</a></i>;<br/>
<a NAME="E8:36"/><b>then </b>
(  <a href="classes1.html#K4" title="CLASSES1:func.4">Rank</a> <span class="p1">(<span class="default"><a href="classes1.html#K6" title="CLASSES1:func.6">the_rank_of</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="classes1.html#K4" title="CLASSES1:func.4">Rank</a> <span class="p1">(<span class="default"><a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c1">W</font></span>)</span> &amp; <font color="Maroon" title="c2">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="classes1.html#K4" title="CLASSES1:func.4">Rank</a> <span class="p1">(<span class="default"><a href="classes1.html#K6" title="CLASSES1:func.6">the_rank_of</a> <font color="Maroon" title="c2">X</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="classes1.html#T42" title="CLASSES1:th.42">CLASSES1:42</a>, <a class="ref" href="classes1.html#D8" title="CLASSES1:def.8">CLASSES1:def 8</a></i>;<br/>
<b>hence </b><a NAME="E9:36"/>
<font color="Maroon" title="c2">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="classes1.html#K4" title="CLASSES1:func.4">Rank</a> <span class="p1">(<span class="default"><a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c1">W</font></span>)</span>
 <b>by </b><i><a class="ref" href="classes1.html#T47" title="CLASSES1:th.47">CLASSES1:47</a></i>;<br/>


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